X(t) := X1(t1) + …+ X_N(t_N) (t ∈ RN+).
Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set X-1({0}) of X to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of X-1({0}) which hold with positive probability in the case that X-1({0}) can be non-void. Here, we prove that the Hausdorff dimension of X-1({0}) is a constant almost surely on the event {X-1 ({0}) ≠ ∅}. Moreover, we derive a formula for the said constant. This portion of our work extends the one-parameter formulas of Horowitz (1968) and Hawkes (1974). More generally, we prove that for every non-random Borel set F in (0,∞)N, the Hausdorff dimension of X-1({0}) ∩ F is a constant almost surely on the event {X-1 ({0}) ∩ F ≠ ∅ }. This constant is computed explicitly in many cases.Keywords. Additive Lévy processes, level sets, Hausdorff dimension
AMS Classification (2000) Primary. 60G70 Secondary. 60F15
Support. Research supported in part by a grant from the National Science Foundation grant DMS-0404729.
Pre/E-Prints. This paper is available in
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Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |
Narn-Rueih Shieh Department of Mathematics, National Taiwan University Taipei 10617, Taiwan shiehnr@math.ntu.edu.tw |
Yimin Xiao Department of Statistics and Probability, A-413 Wells Hall Michigan State University East Lansing, MI 48824, USA xiao@stt.msu.edu |
Last Update: September 25, 2008
© 2006 - Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao